Displacement Measurements, Speed ​​or Acceleration

This article explains why measurements are carried out in displacement, speed or acceleration as part of a vibration monitoring in predictive maintenance. Throughout the text we see how the choice depends on the frequency of the phenomenon, the type of defect and the physical degradation mechanism involved.

The choice between measurements in displacement, velocity or acceleration is not arbitrary. It results from the way in which each of these quantities relates to the vibration frequency and with the physical mechanism responsible for machine degradation.

The three quantities describe exactly the same vibratory movement, but emphasize different frequency ranges and different mechanical effects.

1. Physical relationship between displacement, velocity and acceleration

These quantities are linked by temporal derivatives:

• Speed ​​= derivative of displacement
• Acceleration = derivative of speed

For a sinusoidal vibration:

• displacement: x(t) = X·sin(ωt)

Deriving in order of time, instantaneous speed and acceleration are obtained:

• v(t) = dx/dt = Xω·cos(ωt) → peak amplitude: V = Xω
• a(t) = dv/dt = −Xω²·sin(ωt) → peak amplitude: A = Xω²

That is, the peak amplitudes of each quantity are:

• velocity: V = Xω
• Acceleration: A = Xω²

at where:

• X = displacement (peak amplitude)
• V = speed (peak amplitude)
• A = acceleration (peak amplitude)
• ω = 2πf, where f is the frequency (in Hz)

This results in a fundamental fact:

• speed increases proportionally to frequency;
• acceleration increases with the square of frequency.

Like this, the same vibration may appear small in displacement and very large in acceleration if it occurs at high frequency.

2. The most important factor: the frequency of the phenomenon

Low frequencies

(< 10–20 Hz)

The physical movement of the machine predominates.

It is more appropriate to measure:

• displacement

Examples:

• imbalance of large rotors;
• structural oscillations;
• building vibration;
• hydraulic turbines.

Average frequencies

(10–1000 Hz approximately)

The energy transmitted to the structure is best represented by the velocity.

That is why it is used:

• RMS speed

It is the parameter adopted by ISO standards to evaluate the severity of vibration.

Examples:

• engines;
• bombs;
• fans;
• reducers;
• compressors.

High frequencies

(>1000 Hz)

Displacement amplitudes become extremely small.

However, acceleration grows quickly.

Like this, measure:

• acceleration

Examples:

• bearing defects;
• impacts;
• cavitation;
• gears;
• cracks;
• shocks.

3. The degradation mechanism

The choice also depends on the type of failure.

Type of defect	Preferred magnitude	Justification
Imbalance	Displacement or speed	Large rotor movement
Misalignment	velocity	High vibratory energy
Days off	Speed ​​and acceleration	Movement + impacts
bearings	Acceleration	High frequency impacts
gears	Acceleration	High frequencies
cavitation	Acceleration	hydraulic shocks
Shafts in hydrodynamic bearings	displacement	Relative shaft movement

4. Relationship with mechanical damage

Each quantity is related to a different type of effort.

displacement

Relates to:

• relative motion;
• days off;
• interference;
• contact between components.

Question that answers:

How much the machine moves?

velocity

Relates to:

• vibratory energy;
• transmitted power;
• structural fatigue.

Question that answers:

How much vibratory energy is being transmitted?

That is why it is used by ISO 20816.

Acceleration

Relates to:

• dynamic forces.

By Newton's second law,

F = m · a

greater acceleration implies greater inertia forces.

Question that answers:

What instantaneous forces are acting on the machine?

It is therefore particularly sensitive to impacts and incipient defects.

5. Frequency sensitivity

Consider a constant displacement vibration.

As the frequency increases:

Frequency	displacement	velocity	Acceleration
10 Hz	high	reduced	very reduced
100 Hz	equal	10× largest	100× largest
1000 Hz	equal	100× largest	10 000× largest

Consequently:

• offset “favors” low frequencies;
• speed is approximately balanced over a wide range of frequencies;
• acceleration highlights high frequencies.

Figure 1 – Displacement, speed and acceleration as a function of frequency, for a constant displacement amplitude (log-log scale). Velocity increases proportionally to f and acceleration to f², confirming the behavior described in the table above.

Next you can see the vibration in displacement, speed and acceleration in a real machine.

6. Why speed is used in ISO standards?

Speed ​​constitutes a compromise between displacement and acceleration.

• Does not overvalue low frequencies;
• Does not overvalue high frequencies;
• Correlates well with vibrational energy;
• Correlates well with structural damage in many industrial machines.

Therefore, Most severity assessment standards use mm/s RMS.

7. Because to detect damage to bearings they use acceleration?

When a small defect appears on a raceway:

• the displacement produced may be less than 0,1 μm;
• the speed remains very low;
• but the impact generates accelerations of hundreds of g for a few microseconds.

Acceleration allows you to detect these defects long before they are visible in speed or displacement.

8. Because to API 670 uses displacement?

In turbomachines supported on hydrodynamic bearings, it is important to know the movement of the shaft in relation to the bearing.

The critical parameter is:

• the orbit of the shaft;
• the available slack;
• the possibility of contact between the shaft and the bearing.

In these cases, the displacement is measured with proximity probes, normally in μm peak-to-peak.

9. resume

Greatness	Better frequency range	Phenomena that evidence	Typical Applications
displacement (μm)	Low frequencies	Physical movement and excursion of the rotor	turbines, hydrodynamic bearings, large rotors, orbit analysis
velocity (mm/s RMS)	Average frequencies	Vibrational energy and global severity	engines, bombs, fans, reducers, monitoring according to ISO 20816
Acceleration (m/s² or g)	High frequencies	Dynamic forces, impacts and incipient defects	bearings, gears, cavitation, shocks and early diagnosis

10. Displacement Measurements, Speed ​​or Acceleration – Conclusion

Selecting between displacement, speed and acceleration essentially depends on four factors:

1. The frequency range of the vibratory phenomenon;
2. The type of defect that is intended to identify;
3. The physical mechanism of degradation (movement, energy or strength);
4. The purpose of measurement, whether general monitoring, detailed diagnosis or machine protection.

It is for this reason that modern vibration monitoring and analysis systems often record all three quantities simultaneously., using digital integration or differentiation of the accelerometer signal, to provide the most appropriate information for each type of analysis.

In practice, most of data collectors and analyzers directly measures acceleration with an accelerometer and obtains speed by digital integration of the signal, and displacement by double integration. This process works well, but requires some care: integration amplifies low-frequency noise, therefore it is necessary to apply high-pass filters (high-pass) suitable to avoid drifts (drift) and distortions in the calculated displacement and speed values.

11. References

5. ISO 20816-1:2016 – Mechanical vibration — Measurement and evaluation of machine vibration — Part 1: General guidelines.
6. ISO 20816-3:2022 – Mechanical vibration — Measurement and evaluation of machine vibration — Part 3: Industrial machines with a power rating above 15 kW and operating speeds between 120 r/min and 30 000 rpm.
7. ISO 20816-9:2020 – Mechanical vibration — Measurement and evaluation of machine vibration — Part 9: Gearboxes.
8. API Standard 670, 5th Edition – Machinery Protection Systems, American Petroleum Institute.
9. ISO 13373-1 – Condition monitoring and diagnostics of machines — Vibration condition monitoring — Part 1: General procedures.